The infinite sum 513+55132+555133+5555134+..... equals

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Question

The infinite sum

\frac{5}{13} + \frac{55}{13^{2}} + \frac{555}{13^{3}} + \frac{5555}{13^{4}} + . . . . .

equals

Answer: Option C
:
C

\frac{5}{13} + \frac{55}{13^{2}} + \frac{555}{13^{3}} + \frac{5555}{13^{4}} + . . . . . = \frac{5}{9} (\frac{9}{13} + \frac{99}{13^{2}} + \frac{999}{13^{3}} + . . . . . . . . . . . . .)

= \frac{5}{9} [\frac{10 - 1}{13} + \frac{100 - 1}{13^{2}} + \frac{1000 - 1}{13^{3}} . . . . . . . . . . . . . .]

= \frac{5}{9} [{{(\frac{10}{13})}^{1} + {(\frac{10}{13})}^{2} + {(\frac{10}{13})}^{3} + . . . . . . . . . .} - {(\frac{1}{13}) + {(\frac{1}{13})}^{2} + {(\frac{1}{13})}^{3} + . . . . . . . .}]

Now solve this infinite GP to get

\frac{65}{36}

Hence, option (c)

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